lawn mower for putting green?

okay.my parents have decided to let me get a putting green for my backyard because we have the right soil etc and it will be relitivly in-expensive.but they are not paying hundreds for a putting green mower that will cut it at the correct hight.
i understand that you need a cylinder mower and not a rotary.i have looked at some basic cylinder mowers that are in-expensive but the problem is that they only cut down as far as 10mm.i need it to cut down to 4-6 mm for it to be the correct length.

so.im wondering,if you take the wheels off the mower (which cuts at 10mm)and put on smaller wheels..then it would be lower and cut at a lower hight..would that work?
need feedback.
Thanks!